Multiple choice general knowledge math & puzzles

what is the maximum value of n such that 77! is perfectly divisible by 720^n(720 reaise to power n)?

  1. 35

  2. 18

  3. 17

  4. 36

Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

720 = 2^4 × 3^2. For 720^n to divide 77!, we need at least 4n factors of 2 and 2n factors of 3 and n factors of 5. Count of 5s in 77! is floor(77/5) + floor(77/25) = 15 + 3 = 18. Count of 3s is floor(77/3) + floor(77/9) + floor(77/27) + floor(77/81) = 25 + 8 + 2 + 0 = 35. We need 2n ≤ 35 and n ≤ 18, giving n ≤ 17 (since 2×18 = 36 > 35).